Comment by woolion
1 month ago
From the point of view of proof-theory, you can show that PA (arithmetic) is equivalent to the consistency of the omega cardinal (the countable infinite). Basically, everything line up quite well between things you want to be true and things that are true in that system. This equivalence breaks down with higher-order system such as System F, but it gives a system that may feel more natural, especially to programmers. The problem that explains the endurance of "formalism" is that there are so many things that you "want to be true" that can't be shown to be true in intuitionistic systems is a real issue. For instance, simply proving that a fast-growing function is total. You are fine with recurrence, but not if the function grows too fast? This sounds really stupid. But I don't think many people care that much, they'll just use whatever give them results.
It is certainly easier to prove interesting theorems with formalism. You don't get caught up with such basic things like whether or not it is always possible to tell that one real number is bigger than another.
But formalism leads to having to accept conclusions that some of us don't like. I already referred to the existence of uncountably many things that cannot in any useful way ever be specified. If you include the axiom of choice, you get the Banach-Tarski paradox. Mathematicians debated that one for a while, but now generally accept it.
My favorite example of a weird conclusion comes from https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo.... We can non-constructively prove the following facts. Any class of graphs that is closed under the graph minor operation (for example planar graphs), has a finite set of forbidden graph minors that completely characterize the graph (in the case of planar graphs, K5 and K3,3). In general, there is no way to find those forbidden graph minors. Even if you were given the complete list, you couldn't necessarily verify that the list was correct. You cannot necessarily even find an upper bound on how big this set is.
By "cannot necessarily" I mean, "it is sometimes unprovable".
In what sense can a finite set exist and be finite when it is unfindable, unverifiable, and has unboundable size?
To make this concrete, there are 17,523 known forbidden minors for the toroidal graphs. We don't know how to find more. We don't know if we have the full list. And we don't have an upper bound on how many more of them there are to be found.
You're free to accept this ephemeral claim to existence as actual existence. But this existence isn't very useful for us.
> In what sense can a finite set exist and be finite when it is unfindable, unverifiable, and has unboundable size?
In the same sense that we could say that every computer program must either eventually terminate or never terminate without most people thinking there's a major philosophical problem here.
And by the way, the very same question can be (and has been) levelled at constructivism: in what sense does a result that would take longer than the lifetime of the universe to compute exist, as it is unfindable and unverifiable?
Look, I think that it is interesting to work with constructive axioms, but I don't think that humans philosophically reject non-constructive results. It's one thing to say that we can learn interesting things in constructive mathematics and another to say there's a fundamental problem with non-constructive mathematics.
> But formalism leads to having to accept conclusions that some of us don't like.
At least in Hilbert's sense, I don't think formalism says quite what you claim it says. He says that some mathematical statements or results apply to things we can see in the world and could be assigned meaning through, say, correspondence to physics. But other mathematical statements don't say anything about the physical world, and therefore the question of their "actual meaning" is not reason to reject them as long as they don't lead to "real" results (in the first class of statements) that contradict physical reality.
Formalism, therefore, doesn't require you to accept or reject any particular meaning that the second class of statements may or may not have. If a statement in the second class says that some set exists, you don't have to assign that "existence" any meaning beyond the formula itself.
> Look, I think that it is interesting to work with constructive axioms, but I don't think that humans philosophically reject non-constructive results.
I don't think the point of constructivism is to "philosophically reject non-constructive results". You can accept non-constructive results just fine as a constructivist, so long as they're consistently rephrased as negative statements, i.e. logical statements starting with "NOT ...". This is handy in some ways (you now know instantly what statements correspond to "direct" proofs that can be given a computational semantics and even be reused for all sorts of computer sciencey stuff) and not so handy in others (to some extent, it comes with a kind of denial about the inherently "dual" nature of the fragment of your constructive logic that contains all that negatively-phrased stuff). But these are matters of aesthetics and perceived elegance, more than philosophy.
The duality concern is one that some will want to address by moving even further to linear logics (since these are "dual" like classical logic but also allow for constructive statements) but of course that's yet another can of worms in its own right.
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My understanding of how Hilbert meant it is summed up in this quote from him: "Mathematics is a game played according to certain simple rules with meaningless marks on paper." I think that in part because I read Constance Reid's excellent biography Hilbert! It traces in some detail his thinking over his life, and how he came to formalism. His thinking about the nature of existence was particularly interesting.
If you think that he meant something else, please find somewhere where he said something that didn't boil down to that.
As for what most people think about the philosophical implications, nobody should be expected to have any meaningful philosophical opinions about topics that they have not yet tried to think about. I know that I didn't.
After you've thought about it, you may well have a dramatically different opinion than I do. For example Gödel thought that mathematical existence was real, since mathematics exists in God's mind. This idea made it important to him to decide which set of axioms was right, where right means, "These are the axioms that God must have settled on, and that therefore exist in His mind." This lead to such ironies as the fact that after proving that the consistency of ZF implies the consistency of ZFC, he then concluded that that the construction was so unnatural that Choice couldn't be one of God's axioms!
I don't agree with Gödel. For a start, I don't believe that God exists. And after I thought about it more, I realized that what I want existence to mean, isn't what mathematicians mean when they say "exists". I'm willing to use language in their way when I'm talking to them. But I'm always aware that it doesn't mean what I want it to mean.
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I'm fine with that. I don't think it's much worse than the quirks of what you call non-formalists systems.
In your original comment, you mention:
All notions about uncountable sets being larger than countable ones, require separating the notion of truth from the reasoning required to establish that truth.
If you wanted to formalize something like that, you'd need the consistency of an absurdly large cardinal. I think it is an interesting type of question to explore, so it's fine to have these large cardinals.
I believe that you're fine with it, simply because that is what you're familiar with. And if you'd grown up with a different way of thinking about these problems, then you wouldn't be fine with it.
Personally, I can work with either system. But, to me, formalism really does feel like a game. And the more that I have thought about the foundations of math, the more dissatisfied I have become with this game. And now I find myself frustrated when people assert the conclusions of the game as truth, instead of as merely being formal statements within a game that mathematicians are choosing to play.
Here is something that I believe.
We owe our current understanding of uncountability to Cantor's metaphor, about figuring out which group of sheep is larger by pairing them off. We would today have a very different kind of mathematics if Cantor had instead made a more careful analogy to the problem of trying to count all of the sheep that have ever existed. Even if you had perfect information about the past, you're doomed to fail because you can't figure out where to draw the line between ancestral sheep, and sheep-like ancestors.
This second metaphor is exactly parallel to uncountability within the computable universe. For example we can implement reals as some kind of Cauchy sequences. For example as programs for functions f, where f(n) is a rational, and |f(n) - f(m)| <= 1/n + 1/m. This works perfectly well. But now Cantor's diagonalization argument clearly does not demonstrate that there are more reals. Instead it demonstrates the limits of what computation can predict about the behavior of other computations.
In other words, I just described a system operating on a notion of truth that is directly tied to the reasoning required to establish that truth. And in that system, uncountable is tied to self-reference. And really doesn't mean more.
I don't know how to really formalize this idea. But I'd be interested if anyone actually has done so.
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> In what sense can a finite set exist and be finite when it is unfindable, unverifiable, and has unboundable size?
The way I see it is that an existence of proof isn't required for something to be true. Something being true is a matter of the model, being provable is a matter of axioms and deduction rules. And there comes the distinction between ⊨ and ⊢.